On subgroups which cover Frattini chief factors of a finite group
S. Kamornikov, O. Shemetkova, Yi Xiaolan
All groups considered in the paper are finite. We use
standard definitions and notations from [1–2].
Definition 1. An element \(x\) of a group \(G\)
is called
\(Q\)-superfrattini if it satisfies one of the following two equivalent
conditions:
\(1)\) each chief factor \(A/B\) of \(G\) for which \(x \in A \setminus B \) is
Frattini;
\(2)\) each chief factor of \(G\) having a form \(\langle x^G \rangle/B\) is
Frattini.
By definition, the identity element of \(G\) is \(Q\)-superfrattini.
Definition 2. A subgroup \(H\) of a group \(G\) is called a \(\Phi\)-isolator
if it covers all Frattini chief factors and avoids all non-Frattini
chief factors of \(G\).
A connection between \(Q\)-superfrattini elements of \(G\) and its
\(\Phi\)-isolators is investigated.
Theorem.For a group \(G\), the following conditions hold:
\(1)\) any two \(\Phi\)-isolators of \(G\) have equal order;
\(2)\) if \(H\) is a \(\Phi\)-isolator of \(G\), then:
\(a)\) all elements of \(H\) are \(Q\)-superfrattini in \(G\);
\(b)\) \(H\) is not contained properly in any subgroup of \(G\) whose elements
are all \(Q\)-superfrattini in \(G\);
\(c)\) \(Core_{G}(H)=\Phi(G)\);
\(d)\) if \(H\unlhd G\), then \(H=\Phi(G)\).
In 1962, Gaschütz introduced in [3] the concept of the prefrattini
subgroup of a finite soluble group. In the original presentation, the
prefrattini subgroup is defined as the intersection of
complements of the crowns of all non-Frattini chief factors of a fixed chief series of
the group.
By definition, every
soluble group has at least one prefrattini subgroup. As shown in
[3], each prefrattini subgroup covers all Frattini and avoids all
non-Frattini chief factors of a soluble group \(G\), i.e. it is a
\(\Phi\)-isolator of \(G\).
Proposition 1.Let \(H\) be a prefrattini subgroup of a soluble
group \(G\). Then all elements of \(H\) are \(Q\)-superfrattini in \(G\).
Proposition 2.Let \(H\) be a \(\Phi\)-isolator of a soluble group
\(G\). Then \(H\) is a prefrattini subgroup if and only if
\(H\) permutes with every element of a Hall system of \(G\).
References
L. A. Shemetkov, Formations of Finite Groups, Moscow: Nauka (1978).
K. Doerk, T. O. Hawkes, Finite Soluble Groups, Berlin - New-York:
Walter de Gruyter (1992).
W. Gaschütz, Praefrattinigruppen, Arch. Math., 13, N3, (1962), 418–426.
